Self-images of periodic phase elements in the fractional Fourier transform domain

General conditions of periodic phase elements self-images forming (Talbot effect) in the fractional Fourier transform (FrFT) domain is given. Analytical solution of the FrFT images intensity distribution for the different forms (binary, linear, parabolic and others) of periodic elements low-level cell profiles is presented. Intensity difference Delta I measuring of the FrFT periodic self-image allow to determine the phase difference A(p of periodic elements low-level cell profile. Theory of the FrFT images forming of periodic phase elements based on signal distribution method is given. We use ambiguity function A(ff).(x(0);w(0)) in difference conjugate coordinates (x(0);w(0)) as base functional of the periodic phase element distribution. The FrFT distribution A(upup). (x(0); omega(0)) corresponds to the rotation matrix T,, which describe rotation of the input signal distribution on an angle phi = p pi / 2, p = 0 1 - the FrFT parameter. The signal distribution method allow to obtain general formula of intensity distribution I-p (x) of the periodic phase element FrFT image. Theoretically proved that at condition F. / tan 0 = I, where F. = T-2 / 4 gimel d - Fresnel number, T - phase element period, A - wave-length, d - length, periodic phase elements self-images are forming in the FrFT domain. In this case interference term is written as delta - function and intensity distribution I(x) of the FrFT self-images is forming as superposition of the cross displaced on a quarter of period self-images of neighboring phase low-cells. Analysis of the FrFT self-images forming at condition 2F(0) / tan phi = 1 is also given. The results of numerical calculations of the periodic phase elements self-images at the different values of the FrFT parameter p are presented. Analytical dependence of the FrFT self-images contrast from phase difference Delta(p) is obtained and the questions about phase microrelief parameters restoration of the phase element low-cell are discussed.